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Computer Modeling in Engineering & Sciences
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DOI: 10.32604/cmes.2024.052307

ARTICLE

A Hermitian C2 Differential Reproducing Kernel Interpolation Meshless Method for the 3D Microstructure-Dependent Static Flexural Analysis of Simply Supported and Functionally Graded Microplates

Chih-Ping Wu* and Ruei-Syuan Chang

Department of Civil Engineering, National Cheng Kung University, Tainan, 70101, Taiwan
*Corresponding Author: Chih-Ping Wu. Email: cpwu@mail.ncku.edu.tw
Received: 29 March 2024; Accepted: 21 June 2024; Published: 20 August 2024

Abstract: This work develops a Hermitian C2 differential reproducing kernel interpolation meshless (DRKIM) method within the consistent couple stress theory (CCST) framework to study the three-dimensional (3D) microstructure-dependent static flexural behavior of a functionally graded (FG) microplate subjected to mechanical loads and placed under full simple supports. In the formulation, we select the transverse stress and displacement components and their first- and second-order derivatives as primary variables. Then, we set up the differential reproducing conditions (DRCs) to obtain the shape functions of the Hermitian C2 differential reproducing kernel (DRK) interpolant’s derivatives without using direct differentiation. The interpolant’s shape function is combined with a primitive function that possesses Kronecker delta properties and an enrichment function that constituents DRCs. As a result, the primary variables and their first- and second-order derivatives satisfy the nodal interpolation properties. Subsequently, incorporating our Hermitian C2 DRK interpolant into the strong form of the 3D CCST, we develop a DRKIM method to analyze the FG microplate’s 3D microstructure-dependent static flexural behavior. The Hermitian C2 DRKIM method is confirmed to be accurate and fast in its convergence rate by comparing the solutions it produces with the relevant 3D solutions available in the literature. Finally, the impact of essential factors on the transverse stresses, in-plane stresses, displacements, and couple stresses that are induced in the loaded microplate is examined. These factors include the length-to-thickness ratio, the material length-scale parameter, and the inhomogeneity index, which appear to be significant.

Keywords: Consistent/modified couple stress theory; differential reproducing kernel methods; microplates; point collocation methods; static flexural; 3D microstructure-dependent analysis

1  Introduction

With the increasing demand for microstructures in industry and the rapid progress in material manufacturing technology, functionally graded (FG) structures have gradually shrunk from the macro scale to the micron scale. FG microstructures are gradually being used in cutting-edge technology fields, including thin films [1,2], micro-electro-mechanical systems [3,4], and atomic force microscopes [5,6]. Thus, developing an effective computational method to investigate the mechanical behavior of these microstructures has attracted considerable attention.

It is well-known that the mechanical behavior of FG macrostructures will be changed as their dimensions shrink from the macro-scale to the micro-scale [7]. The existing shell, plate, and beam theories based on the classical continuum mechanics (CCM) are inappropriate for use to analyze the dynamic and static responses of FG microshells, microplates, and microbeams due to the microstructure-dependent effect becoming significant. As a result, some non-CCM-based theoretical methods accounting for the microstructure-dependent impact have been proposed to investigate the mechanical behavior of microstructures. These theoretical methods include the couple stress theory (CST) [8,9], the strain gradient theory (SGT) [10,11], the doublet mechanics theory [12], the micropolar elasticity theory [13,14], and the nonlocal elasticity theory [15].

Hadjesfandiari et al. [16,17] and Yang et al. [18] established the consistent CST (CCST) and the modified CST (MCST) by assuming the couple-stress tensor is skew-symmetric and symmetric, respectively. As a result, instead of two material length-scale coefficients, which are required to study an elastic isotropic body’s mechanical behavior when the original CST is employed, only one material length-scale coefficient is needed when the MCST/CCST is employed. This facilitates their future application.

Within the CCST/MCST framework, some two-dimensional (2D) shear deformation theories for investigating the microstructure-dependent mechanical behavior of FG microplates/microshells have been developed by assuming particular kinematics models a priori. Beni et al. [19] presented a microstructure-dependent classical shell theory on the basis of the MCST to determine an FG circular cylindrical microshell’s smallest natural frequency and its corresponding wave number pair. Incorporating Mindlin’s kinematics model into the MCST, Ma et al. [20] established a microstructure-dependent first-order shear deformation theory (FOSDT) to analyze a homogeneous isotropic microplate’s flexural and free vibration behaviors. Arefi et al. [21] developed a novel shear deformation theory on the basis of the MCST to examine a three-layered microplate’s stress and displacement, for which the microplate of interest consists of an exponentially graded (EG) core and two piezomagnetic face sheets. Based on Hamilton’s principle combined with Reddy’s kinematics model, Lei et al. [22] and Thai et al. [23] presented a microstructure-dependent refined shear deformation theory (RSDT) on the basis of the MCST to conduct an FG microplate’s microstructure-dependent deformation and natural frequency behavior analyses. Kim et al. [24] developed a microstructure-dependent and MCSD-based third-order shear deformation theory (TOSDT) to investigate an FG microplate’s static buckling, static flexural, and free vibration behaviors. Thai et al. [25] developed a microstructure-dependent sinusoidal shear deformation theory (SSDT) on the basis of the MCST to examine an FG microplate’s static flexural and free vibration behaviors. Sobhy et al. [26] presented a microstructure-dependent and MCST-based trigonometric shear deformation theory (TSDT) with four primary variables for modeling an EG microplate’s static buckling, static flexural, and free vibration characteristics resting on Pasternak’s foundation.

Unlike these 2D microstructure-dependent and MCST-based shear deformation theories mentioned above, Wu et al. [27] established the unified microstructure-dependent shear deformation theories based on the CCST to study an FG/EG elastic microplate’s mechanical behavior. Their results showed that the CCST and MCST solutions of deformation and natural frequency associated with out-of-plane vibration modes are almost identical when setting the value of MCST’s material length-scale parameter at twice that of CCST’s material length-scale parameter. However, their solutions of natural frequency associated with the in-plane vibration modes are slightly different.

Instead of the CCST and MCST, other non-CCM-based analytical and numerical methods, including the SGT, the differential quadrature method (DQM), the iso-geometric analysis technique (IGAT), etc., have also been employed to study an FG microplate’s mechanical behavior. Incorporating Kirchhoff-Love’s kinematics model into the SGT, Deng et al. [28] established a non-CCM-based theory to determine an FG microplate’s smallest natural frequency with variable thickness. Within the SGT framework, Balobanov et al. [29] presented a microstructure-dependent classical thin shell theory to investigate a circular cylindrical microscale shell’s static flexural behavior. Integrating the advantages of the finite element method (FEM) and the DQM, Zhang et al. [30] developed a Hermitian C1 four-node quadrilateral element for conducting a moderately thick microplate’s mechanical behavior analysis. Integrating the MCST and the IGAT, Thanh et al. [31] established a seventh-order shear deformation theory for analyzing a porous FG microplate’s microstructure-dependent nonlinear thermal stability behavior. Nguyen et al. [32] developed a computational approach for analyzing an FG microplate’s geometrically nonlinear behavior on the basis of the IGAT and the RSDT. In conjunction with a modified nonlocal CST and the IGAT, Pham et al. [33] conducted an FG microplate’s static flexural and free vibration characteristics analyses, where the microplate rested on an elastic foundation. Based on the CCST, Wu and his colleagues [34,35] established a semi-analytical Hermitian Cn FEM to conduct elastic and piezoelectric microscale plates’/shells’ microstructure-dependent static and dynamic behavior analyses.

Meshless methods have also been employed to investigate a microscale structure’s mechanical behavior. Incorporating Mindlin’s kinematics model and radial basis functions into the MCST, Roque et al. [36] proposed a point collocation method for analyzing a homogeneous isotropic microplate’s static flexural behavior. Incorporating HSDT’s kinematics model into the MCST, Tran et al. [37] and Thai et al. [38] presented a moving Kriging interpolation meshless method to investigate an FG sandwich microplate’s static buckling, static flexural, and free vibration characteristics. Nguyen et al. [39] incorporated a four-variable kinematics model into the MCST to develop a non-uniform rational B-splines (NURBS) meshless method, which was used to investigate an FG microplate’s microstructure-dependent static buckling, static flexural, and free vibration behaviors. Finally, Thai et al. [40] employed the NURBS meshless method to conduct an FG microplate’s static buckling and free vibration behavior analyses.

In their series of papers, Li et al. [41], Simkins et al. [42], Liu et al. [43], and Lu et al. [44] established the reproducing kernel element method to solve Galerkin weak forms of a system of higher order partial differential equations which are associated with Dirichlet boundary conditions.

Chen et al. [45] and Wang et al. [46] established the Hermitian C1 and Lagrange C0 differential reproducing kernel interpolation meshless (DRKIM) methods, respectively, for investigating laminated composite and FG macroscale structures’ mechanical behavior. The novelty of these DRKIM methods is that the shape functions of the differential reproducing kernel (DRK) interpolant’s derivatives are obtained by setting up the differential reproducing conditions (DRCs) without using direct differentiation, as is necessary for the conventional reproducing kernel interpolation and approximation methods [47]. It has been shown that the solutions obtained using these DRKIM methods closely agree with the available 3D solutions of macroscale plates, rather than those of microplates.

As we see in the strong form of the 3D CCST, the primary variables’ highest order derivative is the third order for microplates, which differs from the first order designation for macroscale plates. This situation will reduce the accuracy of the early proposed Hermitian C1 and Lagrangian C0 DRKIM methods and slow down their convergence rate. To enhance these DRKIM methods’ accuracy and speed up their convergence rate, in this paper, we aim to establish a Hermitian C2 DRKIM method by making some modifications: the Hermitian C2 DRK interpolant should satisfy the nodal interpolation properties and the continuity conditions up to primary variables’ second-order derivatives at each sampling node. Moreover, we also aim to establish the Hermitian C2 DRKIM method, which is a point collocation method, by incorporating our Hermitian C2 DRK interpolant into the strong form of the 3D CCST to carry out an FG microplate’s 3D microstructure-dependent static flexural analysis. After validating the Hermitian C2 DRKIM’s accuracy using the relevant 3D solutions reported in the literature, we will carry out a parametric study for the FG microplate’s 3D microstructure-dependent static flexural behavior to examine how the impact of essential factors affects the induced deformations, in-plane stresses, transverse stresses, and couple stresses, including the length-to-thickness ratio, the material length-scale parameter, and the inhomogeneity index.

2  The Hermitian C2 DRKIM Method

2.1 The Hermitian C2 DRK Interpolant

We consider np discrete sampling nodes placed at ξ = ξ1,  ξ2, ⋯,  ξnp, respectively, in a natural coordinate system ξ, the values of which are ξ1=−1, ξnp=1, and the others are randomly selected between −1 and 1. The Hermitian C2 DRK interpolant fh(ξ) and its first- and second-order derivatives, dfh(ξ)/dξ and d2fh(ξ)/dξ2 (i.e., θh(ξ) and κh(ξ)), are required to satisfy the nodal interpolation properties. Thus, it is defined as

fh(ξ)=∑l=1np  [Nl(ξ)fl+N^l(ξ)θ l+N¯l(ξ) κl]=∑l=1np { [ϕl(ξ)+ψl(ξ)]fl+[ϕ^l(ξ)+ψ^l(ξ)]θl+[ϕ¯l(ξ)+ψ¯l(ξ)]κl},  (1)

where Nl(ξ), N^l(ξ), and N¯l(ξ) (l = 1, 2,…, np) denote the Hermitian C2 DRK interpolant’s shape functions at ξ = ξl; fl, θl, and κl are the nodal values of fh(ξ), θh(ξ), and κh(ξ) at ξ=ξl, respectively. ψl(ξ), ψ^l(ξ), and ψ¯l(ξ) (l = 1, 2,…, np) are the primitive functions for fh(ξ), θh(ξ), and κh(ξ), respectively, which are selected to satisfy the Kronecker delta properties. The primitive functions chosen in this article are ψl(ξ)=wq(ξ), ψ^l(ξ)=(ξ−ξl)wq(ξ), and ψ¯l(ξ)=(ξ−ξl)2wq(ξ)/2, in which wq(ξ) is defined as a normalized eighth-degree (octic) polynomial with the support size a0=(0.99)min(|ξl−ξl+1|,|ξl−ξl−1|), such that these primitive functions and their first- and second-derivatives satisfy the Kronecker delta properties (i.e., ψl(ξk)=δlk, dψ^l(ξk)/dξ=δlk, d2ψ¯l(ξk)/dξ2=δlk, ψ^l(ξk)=ψ¯l(ξk)=0, dψl(ξk)/dξ=dψ¯l(ξk)/dξ=0, and d2ψl(ξk)/dξ2=d2ψ^l(ξk)/dξ2=0). The symbols ϕl(ξ), ϕ^l(ξ), and ϕ¯l(ξ) (l = 1, 2,…, np) are defined as the enrichment functions for fh(ξ), θh(ξ), and κh(ξ), respectively, which are determined by setting up the nth-order DRCs. In our Hermitian C2 DRK interpolant, the enrichment functions are arranged as ϕl(ξ)=wa(ξ−ξl)PT(ξ−ξl)b0c2(ξ), ϕ^l(ξ)=wa(ξ−ξl)P^T(ξ−ξl) b0c2(ξ), and ϕ¯l(ξ)=wa(ξ−ξl)P¯T(ξ−ξl) b0c2(ξ), in which b0c2(ξ) and wa(ξ−ξl) denote the undetermined function vector and a Gaussian function, respectively, and

PT(ξ−ξl)={1(ξ−ξl)(ξ−ξl)2⋯(ξ−ξl)n},(2)

P^T(ξ−ξl)=(−1)dPT(ξ−ξl)/d(ξ−ξl)=dPT(ξ−ξl)/dξl={0,−1,−2(ξ−ξl),−3(ξ−ξl)2,⋯,−n(ξ−ξl)n−1},(3)

P¯T(ξ−ξl)=(−1)2d2PT(ξ−ξ)/d(ξ−ξl)2=d2PT(ξ−ξl)/dξl2={0,0,2,6(ξ−ξl),⋯,n(n−1)(ξ−ξl)n−2}.(4)

In order to determine the undetermined function vector b0c1(z), we select the complete nth-order polynomials as the basis functions that are to be reproduced and set up (n + 1) DRCs as follows:

∑l=1np{[ϕl(ξ)+ψl(ξ)]ξlm+[ϕ^l(ξ)+ψ^l(ξ)]m ξlm−1+[ϕ¯l(ξ)+ψ¯l(ξ)]m (m−1)ξlm−2}=ξmm=0,1, ⋯,n.(5)

Eq. (5) can be rearranged in the following explicit forms:

m=0:∑l=1np[ϕl(ξ)+ψl(ξ)]=1⇒∑l=1np ϕl(ξ)=1−∑l=1np ψl(ξ),(6)

m=1: ∑l=1np[ϕl(ξ)+ψl(ξ)]ξl+∑l=1np[ϕ^l(ξ)+ψ^l(ξ)]=ξ⇒∑l=1np (ξ−ξl) ϕl(ξ)+∑l=1np(−1)ϕ^l(ξ)=−∑l=1np (ξ−ξl) ψl(ξ)−∑l=1np(−1)ψ^l(ξ),(7)

m=2∑l=1np[ϕl(ξ)+ψl(ξ)]ξl2+∑l=1np[ϕ^l(ξ)+ψ^l(ξ)](2ξl) +∑l=1np[ϕ¯l(ξ)+ψ¯l(ξ)](2)=ξ2⇒∑l=1np (ξ−ξl)2ϕl(ξ)+∑l=1np(−2)(ξ−ξl)  ϕ^l(ξ)+∑l=1np2ϕ¯l(ξ)=−∑l=1np (ξ−ξl)2ψl(ξ)−∑l=1np(−2)(ξ−ξl)  ψ^l(ξ)−∑l=1np2ψ¯l(ξ),(8)

⋮

m=n:∑l=1np[ϕl(ξ)+ψl(ξ)]ξln+∑l=1np[ϕ^l(ξ)+ψ^l(ξ)](n ξln−1)+∑l=1np [ϕ¯l(ξ)+ψ¯l(ξ)][n(n−1) ξ ln−2]=ξn⇒∑l=1np (ξ−ξl)n ϕl(ξ)+∑l=1np(−n) (ξ−ξl)n−1ϕ^l(ξ)+∑l=1np(n) (n−1)(ξ−ξl)n−2ϕ¯l(ξ)=−∑l=1np (ξ−ξl)n ψl(ξ)−∑l=1np(−n) (ξ−ξl)n−1ψ^l(ξ)−∑l=1np(n) (n−1)(ξ−ξl)n−2ψ¯l(ξ).(9)

We rewrite Eqs. (6)–(9) in matrix form as follows:

∑l=1np P(ξ−ξl) ϕl (ξ)+∑l=1np P^(ξ−ξl)  ϕ^l (ξ)+∑l=1np P¯(ξ−ξl)  ϕ¯l (ξ)=P(0)−∑l=1np P(ξ−ξl)  ψl (ξ)−∑l=1np P^(ξ−ξl)  ψ^l (ξ)−∑l=1np P¯(ξ−ξl)  ψ¯l (ξ),(10)

where P(0)=[100⋯0]T.

We substitute the enrichment functions into the DRCs to yield the following expression for the undetermined function vector b0c2(ξ):

b0c2(ξ)=Ac2−1(ξ)[P(0)−∑l=1np P(ξ−ξl)  ψl (ξ)−∑l=1np P^(ξ−ξl)  ψ^l (ξ)−∑l=1np P¯(ξ−ξl)  ψ¯l (ξ)],(11)

where Ac2(ξ)=∑l=1np [P(ξ−ξl) wa(ξ−ξl) PT(ξ−ξl)+P^(ξ−ξl) wa(ξ−ξl) P^T(ξ−ξl)+P¯(ξ−ξl)wa(ξ−ξl) P¯T(ξ−ξl)].

We substitute Eq. (11) into Eq. (1) to yield the shape functions of the Hermitrian C2 DRK interpolant as follows:

Nl(ξ)=ϕl(ξ)+ψl (ξ)(l=1,2,…,np),(12)

N^l(ξ)=ϕ^l(ξ)+ψ^l(ξ)(l=1,2,…,np),(13)

N¯l(ξ)=ϕ¯l(ξ)+ψ¯l(ξ)(l=1,2,…,np),(14)

where ϕl(ξ)=wa(ξ−ξl)PT(ξ−ξl)b0c2(ξ),

ϕ^l(ξ)=wa(ξ−ξl)P^T(ξ−ξl)b0c2(ξ),

ϕ¯l(ξ)=wa(ξ−ξl)P¯T(ξ−ξl)b0c2(ξ).

Eq. (11) shows that the enrichment functions vanish at all the sampling points (i.e., ϕl (ξk)=ϕ^l (ξk)=ϕ¯l (ξk)=0, for all l and k = 1, 2, …, np). When we select the primitive functions mentioned above for fh(ξ), such that ψl (ξk)=δlk, ψ^l (ξk)=0, and ψ¯l (ξk)=0 a priori, the shape functions will satisfy the Kronecher delta properties, which are Nl (ξk)=δlk, N^l (ξk)=0, and N¯l (ξk)=0

2.2 The Hermitian C2 DRK Interpolant’s Derivatives

The Hermitian C2 DRK interpolant fh(ξ) in Eq. (1) has the first-order derivative with respect to ξ:

dfh(ξ)dξ=∑l=1np  [Nl(1)(ξ)fl+N^l(1)(ξ)θl+N¯l(1)(ξ)κl]=∑l=1np [ (ϕl(1)(ξ)+ψl(1)(ξ))  fl+(ϕ^l(1)(ξ)+ψ^l(1)(ξ))  θl+ (ϕ¯l(1)(ξ)+ψ¯l(1)(ξ))  κl] ,(15)

where Nl(1)(ξ), N^l(1)(ξ), and N¯l(1)(ξ) (l = 1, 2,…, np) are the shape functions of the Hermitian C2 DRK interpolant’s first-order derivative at the node ξ = ξl, which satisfy the Kronecker delta properties; ψl(1)(ξ), ψ^l(1)(ξ), and ψ¯l(1)(ξ) (l = 1, 2,…, np) are primitive functions’ first-order derivatives (i.e., ψl(1)(ξ)=d ψl(ξ)/dξ, ψ^l(1)(ξ)=d ψ^l(ξ)/dξ, and ψ¯l(1)(ξ)=d ψ¯l(ξ)/dξ); and ϕl(1)(ξ), ϕ^l(1)(ξ), and ϕ¯l(1)(ξ) (l = 1, 2,…, np) denote enrichment functions’ first-order derivatives, which are obtained by imposing the nth-order DRCs, and are expressed as ϕl(1)(ξ)=wa(ξ−ξl)PT(ξ−ξl) b1c2(ξ), ϕ^l(1)(ξ)=wa(ξ−ξl)P^T(ξ−ξl) b1c2(ξ), and ϕ¯l(1)(ξ)=wa(ξ−ξl)P¯T(ξ−ξl) b1c2(ξ), for which b1c2(ξ) is the undetermined function vector.

In order to determine the undetermined functions b1c2(ξ) in Eq. (15), again, we select the complete nth-order polynomials as the basis functions to be reproduced and set up (n + 1) DRCs as follows:

∑l=1np {[ϕl(1)(ξ)+ψl(1)(ξ)]ξlm+[ϕ^l (1)(ξ)+ψ^l(1)(ξ)]m ξlm−1+[ϕ¯l (1)(ξ)+ψ¯l(1)(ξ)]m (m−1)ξlm−2}=m ξm−1,(16)

where m=0,1,2,  ⋯,  n.

We rearrange Eq. (16) in the explicit forms as follows:

m=0:∑l=1np [ϕl(1)(ξ)+ψl(1)(ξ)]=0⇒∑l=1np ϕl(1)(ξ)=−∑l=1np ψl(1)(ξ),(17)

m=1:∑l=1np [ϕl(1)(ξ)+ψl(1)(ξ)]  ξl+∑l=1np [ϕ^l(1)(ξ)+ψ^l(1)(ξ)]=1⇒∑l=1np (ξ−ξl)ϕl(1)(ξ)+∑l=1np (−1)ϕ^l(1)(ξ)=−1−∑l=1np (ξ−ξl)ψl(1)(ξ)−∑l=1np (−1)ψ^l(1)(ξ),(18)

m=2:∑l=1np [ϕl(1)(ξ)+ψl(1)(ξ)]  ξl2+∑l=1np [ϕ^l(1)(ξ)+ψ^l(1)(ξ)]  (2ξl)+∑l=1np [ϕ¯l(1)(ξ)+ψ¯l(1)(ξ)]  (2)=2ξ⇒∑l=1np (ξ−ξl)2ϕl(1)(ξ)+∑l=1np (−2)(ξ−ξl)  ϕ^l(1)(ξ)+∑l=1np (2)ϕ¯l(1)(ξ)=−∑l=1np (ξ−ξl)2ψl(1)(ξ)−∑l=1np (−2)(ξ−ξl) ψ^l(1)(ξ)−∑l=1np (2)ψ¯l(1)(ξ),(19)

⋮

m=n:∑l=1np [ϕl(1)(ξ)+ψl(1)(ξ)]  ξln+∑l=1np [ϕ^l(1)(ξ)+ψ^l(1)(ξ)]  n ξln−1+∑l=1np [ϕ¯l(1)(ξ)+ψ¯l(1)(ξ)]  n (n−1)ξln−2=n ξn−1⇒∑l=1np (ξ−ξl)nϕl(1)(ξ)+∑l=1np (−n) (ξ−ξl)n−1ϕ^l(1)(ξ)+∑l=1np (n) (n−1)(ξ−ξl)n−2ϕ¯l(1)(ξ)=−∑l=1np (ξ−ξl)nψl(1)(ξ)−∑l=1np (−n)(ξ−ξl)n−1ψ^l(1)(ξ)−∑l=1np n (n−1)(ξ−ξl)n−2ψ¯l(1)(ξ).(20)

We rewrite the above Eqs. (17)–(20) in matrix form as follows:

∑l=1np P(ξ−ξl)  ϕ l(1)(ξ)+∑l=1np P^(ξ−ξl)  ϕ^l(1)(ξ)+∑l=1np P¯(ξ−ξl)  ϕ¯l(1)(ξ)=P^(0)−∑l=1np P(ξ−ξl)  ψl(1)(ξ)−∑l=1np P^(ξ−ξl)  ψ^l(1)(ξ)−∑l=1np P¯(ξ−ξl)  ψ¯l(1)(ξ),(21)

where P^(0)=dP(0)/dξl=[0−10⋯0]T.

We substitute the enrichment functions into the DRCs to yield the undetermined function vector b1c2(ξ) as follows:

b1c2(ξ)=Ac2−1(ξ)[P^(0)−∑l=1np P(ξ−ξl)  ψl(1)(ξ)−∑l=1np P^(ξ−ξl)  ψ^l(1)(ξ)−∑l=1np P¯(ξ−ξl)  ψ¯l(1)(ξ)].(22)

We substitute Eq. (22) into Eq. (15) to obtain the shape functions of the Hermitian C2 DRK interpolant’s first-order derivatives as follows:

Nl(1)(ξ)=ϕ l (1)(ξ)+ψ l (1)(ξ)(l=1,2,…,np),(23)

N^l(1)(ξ)=ϕ^l (1)(ξ)+ψ^l (1)(ξ)(l=1,2,…,np),(24)

N¯l(1)(ξ)=ϕ¯l (1)(ξ)+ψ¯l (1)(ξ)(l=1,2,…,np),(25)

where ϕ l(1)(ξ)=wa(ξ−ξl)PT(ξ−ξl)b1c2(ξ), ϕ^l(1)(ξ)=wa(ξ−ξl)P^T(ξ−ξl)b1c2(ξ), ϕ¯l(1)(ξ)=wa(ξ−ξl)P¯T(ξ−ξl)b1c2(ξ).

From Eqs. (23)–(25), it can be seen that the values of the enrichment functions’ first-order derivatives at all sampling nodes are zero (i.e., ϕ l(1)(ξk)=ϕ^l(1)(ξk)=ϕ¯l(1)(ξk)=0). Subsequently, suppose we select the first-order primitive functions for dfh(ξ)/dξ such that ψl(1)(ξk)=0, ψ^l(1)(ξk)=δlk, and ψ¯l(1)(ξk)=0, a priori. Finally, the above shape functions satisfy the Kronecker delta properties (i.e., Nl(1)(ξk)=0, N^l(1)(ξk)=δlk, and N¯l(1)(ξk)=0).

Similarly, the above derivation procedure can proceed to the rth-order derivative of the Hermitian C2 DRK interpolant fh (ξ), which is thus expressed in Appendix A.

2.3 Weight Functions and Primitive Functions

In implementing our Hermitian C2 DRKIM method, we must select the weight function and the primitive function in advance. This work uses the normalized Gaussian function as the weight function, which is expressed as follows [47]:

Normalized Gaussian function: wa(s)={e−(s/α)2−e−(1/α)21−e−(1/α)2fors≤10fors>1,(26)

where s=|ξ−ξl|/al, in which al denotes the support size at the reference sampling point l, and the value of α is set at α=0.3.

As mentioned above, we define the primitive functions for the Hermitian C2 DRK interpolant as ψl(ξ)=wq(ξ), ψ^l(ξ)=(ξ−ξl)wq(ξ), and ψ¯l(ξ)=(ξ−ξl)2wq(ξ)/2, respectively, for which wq(ξ) is a normalized eighth-degree (octic) polynomial, which is given as follows [47]:

wq(s)={−3s8+8s6−6s4+1fors≤10fors>1,(27)

where s=|ξ−ξl|/a0, in which a0 is defined as a0=(0.99)min(|ξl−ξl+1|,|ξl−ξl−1|) to ensure the Kronecker delta properties are satisfied (i.e., ψl(ξ=ξk)=δlk, dψ^l(ξk)/dξ=δlk, and d2ψ¯l(ξk)/dξ2=δlk).

It is noticed that for a meshless method, the support size al for the selected weight function wa(ξ) will not be a very small value, often resulting in numerical errors; whereas, it also has to be small enough to preserve the meshless method’s local character due to an increase in the support size also resulting in numerical errors. Chen et al. [45] and Wang et al. [46] thus recommended a compromise range of the value of al to ensure the Hermitian C1 and Lagrange C0 DRKIM methods’ accuracy and convergence rate. It has been recommended as follows: In the case of a uniform sampling node distribution, the appropriate value of al is al=(n+0.1)Δξ, where  Δξ denotes the spacing between the adjacent nodes, and the value of al is constant for each node. In the case of a randomly scattered node distribution, the appropriate value of al is selected to include (2n + 1) nodes and the value of al is variable for each node. This guidance is adopted in this paper.

To have a clear picture related to how the values of these shape functions vary in the natural coordinate, in Fig. 1a–c, we consider a case of 11 sampling nodes with uniform spacing and present the distributions of the enrichment function (ψ6(ξ)), the primitive function (ϕ6(ξ)), and the shape function (N6(ξ)) of node 6 along the natural coordinate axis, respectively, for which N6(ξ)=ϕ6(ξ)+ψ6(ξ). It can be seen in Fig. 1a–c that the Kronecker delta properties, ψ6(ξi)=δi6  and  N6(ξi)=δi6(i=1−11), are satisfied, and ϕ6(ξi)=0(i=1−11). Furthermore, in Fig. 2, we present the distribution of each sampling node’s shape function along the natural coordinate axis, i.e., Ni(ξ)(i=1−11). Again, each shape function is shown to satisfy the Kronecker delta properties and localize in a region of the support size.

Figure 1: Distributions of (a) the enrichment function, (b) the primitive function, and (c) the shape function of node 6 in the natural coordinate in the case of np = 11 with uniform spacing

Figure 2: Distributions of the shape functions (a) (i = 1–3), (b) (i = 4–6), (c) (i = 7–9), and (d) (i = 10 and 11), in the natural coordinate in the case of np = 11 with uniform spacing

3  3D Microstructure-Dependent Static Flexural Analysis of FG/EG Microplates

3.1 The Quasi-State Space Equations of the CCST

This work considers the 3D microstructure-dependent static flexural problem of a simply-supported FG microplate under either a sinusoidally distributed load or a uniform load, and the former loading case is shown in Fig. 3. The symbols h, Lx, and Ly represent the microplate’s height, length, and width, respectively. A Cartesian coordinate system (x, y and z) is oriented so that the xy-plane is the microplate’s mid-plane.

Figure 3: An FG microplate of interest that is subjected to a sinusoidally distributed load

The displacement vector u of the deformed microplate is expressed as u=ux i+uy j+uz k, where i, j, and k represent the unit basis vectors in the x, y, and z directions, respectively.

The strain tensor ε is symmetric, and its relationships with the displacement tensor in Cartesian coordinates are expressed as

εxx=ux,x,(28a)

εyy=uy,y,(28b)

εzz=uz,z,(28c)

γxz=2εxz=ux,z+uz,x,(28d)

γyz=2εyz=uy,z+uz,y,(28e)

γxy=2εxy=ux,y+uy,x,(28f)

where the commas represent the partial derivative of the suffix variable.

The rotation tensor θ is skew-symmetric, and its relationship with the displacement tensor in Cartesian coordinates is expressed as follows:

θx=θzy=(1/2)(uz,y−uy,z),(29a)

θy=θxz=(1/2)(ux,z−uz,x),(29b)

θz=θyx=(1/2)(uy,x−ux,y).(29c)

The symmetric part of the curvature tensor χ-the rotation tensor θ relationship in Cartesian coordinates is expressed as follows:

χxx=θx,x,(30a)

χyy=θy,y,(30b)

χzz=θz,z,(30c)

χxz=(1/2)(θx,z+θz,x),(30d)

χyz=(1/2)(θy,z+θz,y),(30e)

χxy=(1/2)(θx,y+θy,x).(30f)

The skew-symmetric part of the curvature tensor κ-the rotation tensor θ relationship in Cartesian coordinates is expressed as follows:

κx=κzy=(1/2)(θz,y−θy,z),(31a)

κy=κxz=(1/2)(θx,z−θz,x),(31b)

κz=κyx=(1/2)(θy,x−θx,y).(31c)

Hadjesfandiari and Dargush [16,17] indicated that in general, the force-stress tensor (σij) induced in the loaded microplate is asymmetric. Therefore, they separated it into a skew-symmetric part (σ[ij]) and a symmetric (σ(ij)) part, and represented these two parts using brackets and parentheses that surround a pair of indices, respectively. Subsequently, Hadjesfandiari et al. employed the principle of virtual displacements to deduce a result that the couple-stress tensor μ is skew-symmetric, such that μx=μ zy=−μyz, μy=μ xz=−μzx, and μz=μ yx=−μxy. In addition, they also deduced the force-stress tensor’s skew-symmetric part-the couple stress tensor relationship as follows:

σ[ji]=−μ[i, j]=−(1/2)(μi,j−μj,i).(32)

The linear constitutive equations for a loaded orthotropic material microplate are given by

 {σ(xx)σ(yy)σ(zz)σ(yz)σ(xz)σ(xy)}=[c11c12c13000c12c22c23000c13c23c33000000c44000000c55000000c66]{εxxεyyεzzγyzγxzγxy},(33)

 {μxμyμz}=(−8l2)[G32000G13000G21]{κxκyκz},(34)

where cij (i, j = 1–6) are the elastic coefficients; and G32,  G13,  and  G21 are the shear modulus associated with the zy-, xz-, and yx-planes, respectively. The symbol l represents the microplate’s material length-scale parameter, the determination of which refers to Tang et al. [48] and Song et al. [49].

The stress equilibrium equations of a microplate, following Hadjesfandiari and Dargush’s analysis [16,17], are given by

σ(xx),x+σ(yx),y+σzx,z+σ[yx],y=0,(35)

σ(xy),x+σ(yy),y+σzy,z+σ[xy],x=0,(36)

σxz,x+σyz,y+σzz,z=0,(37)

where as mentioned above, σij=σ(ij)+σ[ij] and σ[ij]=−σ[ji]   for i≠j, and σkk=σ(kk).

We employ the direct elimination to reduce the above equations to six partial differential equations which are expressed in terms of six primary variables: three transverse stresses (σzx,  σzy,  and  σzz) and three displacements (ux,  uy,  and  uz).

Substituting Eqs. (32) and (34) into the fifth equation of Eq. (33) leads to

ux,z=−uz,x+c55−1σzx+l2(G21/c55)(ux,xxz+uy,xyz−uz,xxx−uz,xyy)+l2(G32/c55)(ux,yyz+ux,zzz−uy,xyz−uz,xzz)+l2(G32,z/c55)(ux,yy+ux,zz−uy,xy−uz,xz).(38)

Substituting Eqs. (32) and (34) into the fourth equation of Eq. (33) leads to

uy,z=−uz,y+c44−1σzy+l2(G21/c44)(ux,xyz+uy,yyz−uz,xxy−uz,yyy)+l2(G13/c44)(−ux,xyz+uy,xxz+uy,zzz−uz,yzz)+l2(G13,z/c44)(−ux,xy+uy,xx+uy,zz−uz,yz).(39)

Substituting Eq. (28c) into the third equation of Eq. (33) leads to

uz,z=−c~13ux,x−c~23uy,y+c33−1σ(zz),(40)

where c~k3=ck3/c33(k=1  and  2).

Using the relationships of σxz=σzx+2σ[xz] and σyz=σzy+2σ[yz] and Eqs. (29), (31), and (32), we can rewrite Eqs. (35)–(37) as follows:

σzx,z=−σ(xx),x−σ(yx),y−(1/2)μy,xy−(1/2)μx,yy=−Q11ux,xx−Q12uy,xy−c~13σzz,x−c66ux,yy−c66uy,xy+l2G13[ux,xxyy−uy,xxxy−uy,xyzz+uz,xyyz]+l2G32[ux,yyyy+ux,yyzz−uy,xyyy−uz,xyyz],(41)

σzy,z=−σ(xy),x−σ(yy),y−(1/2)μx,xy−(1/2)μy,xx=−Q12ux,xy−Q22uy,yy−c~23σzz,y−c66ux,xy−c66uy,xx+l2G32[−ux,xyyy−ux,xyzz+uy,xxyy+uz,xxyz]+l2G13[−ux,xxxy+uy,xxxx+uy,xxzz−uz,xxyz],(42)

σzz,z=−σzx,x−σzy,y−μx,xz+μz,xx−μy,yz+μz,yy=−σzx,x−σzy,y+2l2G32[−ux,xyyz−ux,xzzz+uy,xxyz+uz,xxzz]+2l2G13[ux,xyyz−uy,xxyz−uy,yzzz+uz,yyzz]+2l2G21[−ux,xxxz−ux,xyyz−uy,xxyz−uy,yyyz+uz,xxxx+2uz,xxyy+uz,yyyy]  +2l2G32,z[−ux,xyy−ux,xzz+uy,xxy+uz,xxz]+2l2G13,z[ux,xyy−uy,xxy−uy,yzz+uz,yyz],(43)

where Qij=cij−(ci3cj3/c33)(i,  j=1  and 2).

Eqs. (38)–(43) represent the quasi-state space equations for the FG microplates’ 3D microstructure-dependent static flexural behavior. In addition, we can reduce these equations for examining FG microscale plates to those for examining FG macroscale plates by letting the value of l zero.

The microplate’s surface and edge boundary conditions are specified in the following forms [16,17]:

On the top and bottom surfaces,

{σzxσzy  σzz  μx  μy}={00q¯z±00}onz=±h/2,(44)

where the positive directions of q¯z− and q¯z+, following the conventions of the 3D elasticity theory, are defined to be downward and upward, respectively.

For simply supported boundary edges, we express the edge boundary conditions in the following forms:

At the edges x=0 and x=Lx,

σxx=u y=u z=μy=μz=0.(45a)

At the edges y=0 and y=Ly,

σyy=u x=uz=μx=μz=0.(45b)

3.2 Fourier Series Expansion Method

This work lets the external loads q¯z−(x, y)=0 and expands q¯z+(x, y) as a double Fourier series as follows:

q¯z+(x, y)=∑m^=1∞∑n^=1∞qm^n^sin⁡m~x sin⁡n~y,(46)

where the symbols m~=m^π/Lx and n~=n^π/Ly; and the symbols m^  and  n^ are the half-wave numbers.

We also express these primary variables as the following double Fourier series:

ux(x,  y,  z)=∑m^=1∞∑n^=1∞um^n^(z)cos⁡m~x sin⁡n~y,(47)

uy(x,  y,  z)=∑m^=1∞∑n^=1∞vm^n^(z)sin⁡m~x cos⁡n~y,(48)

uz(x,  y,  z)=∑m^=1∞∑n^=1∞wm^n^(z)sin⁡m~x sin⁡n~y,(49)

σzx(x,  y,  z)=∑m^=1∞∑n^=1∞σ31m^n^(z)cos⁡m~x sin⁡n~y,(50)

σzy(x,  y,  z)=∑m^=1∞∑n^=1∞σ32m^n^(z)sin⁡m~x cos⁡n~y,(51)

σzz(x,  y,  z)=∑m^=1∞∑n^=1∞σ33m^n^(z)sin⁡m~x sin⁡n~y.(52)

Substituting Eqs. (47)–(52) into the quasi-state space Eqs. (38)–(43) yields

um^n^,z=−m~wm^n^+c55−1σ31m^n^+l2(G21/c55)(−m~2um^n^,z−m~n~vm^n^,z+m~3wm^n^+m~n~2wm^n^)+l2(G32/c55)(−n~2um^n^,z+um^n^,zzz+m~n~vm^n^,z−m~wm^n^,zz)+l2(G32,z/c55)(−n~2um^n^+um^n^,zz+m~n~vm^n^−m~wm^n^,z),(53)

vm^n^,z=−n~wm^n^+c44−1σ32m^n^+l2(G21/c44)[−m~n~um^n^,z−n~2vm^n^,z+(m~2n~+n~3)wm^n^]+l2(G13/c44)(m~n~um^n^,z−m~2vm^n^,z+vm^n^,zzz−n~wm^n^,zz)+l2(G13,z/c44)(m~n~um^n^−m~2vm^n^+vm^n^,zz−n~wm^n^,z),(54)

wm^n^,z=m~c~13um^n^+n~c~23vm^n^+c33−1σ33m^n^,(55)

σ31m^n^,z=m~2Q11um^n^+m~n~Q12vm^n^−m~c~13σ33m^n^+n~2c66um^n^+m~n~c66vm^n^+l2G13(m~2n~2um^n^−m~3n~vm^n^+m~n~vm^n^,zz−m~n~2wm^n^,z)+l2G32(n~4um^n^−n~2um^n^,zz−m~n~3vm^n^+m~n~2wm^n^,z),(56)

σ32m^n^,z=m~n~Q12um^n^+n~2Q22vm^n^−n~c~23σ33m^n^+m~n~c66um^n^+m~2c66vm^n^+l2G32[−m~n~3um^n^+m~n~um^n^,zz+m~2n~2vm^n^−m~2n~wm^n^,z]+l2G13[−m~3n~um^n^+m~4vm^n^−m~2vm^n^,zz+m~2n~wm^n^,z],(57)

σ33m^n^,z=m~σ31m^n^+n~σ32m^n^+2l2G32(−m~n~2um^n^,z+m~um^n^,zzz+m~2n~vm^n^,z−m~2wm^n^,zz) +2l2G13(m~n~2um^n^,z−m~2n~vm^n^,z+n~vm^n^,zzz−n~2wm^n^,zz) +2l2G21[−(m~3+m~n~2)um^n^,z−(m~2n~+n~3)vm^n^,z+(m~4+2m~2n~2+n~4)wm^n^] +2l2G32,z(−m~n~2um^n^+m~um^n^,zz+m~2n~vm^n^−m~2wm^n^,z) +2l2G13,z(m~n~2um^n^−m~2n~vm^n^+n~vm^n^,zz−n~2wm^n^,z).(58)

3.3 The Hermitian C2 DRKIM Method

This section develops the Hermitian C2 DRKIM method, which is a point collocation, for solving the strong form of the 3D CCST, which is composed of the quasi-state space Eqs. (53)–(58) and their associated boundary conditions (45a) and (45b).

First, we select nc collocation points in the thickness direction, for which nc = 3np, and then substitute the primary variables expressed in Eq. (1) and their relevant derivatives into the quasi-state space Eqs. (53)–(58) at the ith-collocation point, which leads to the following algebraic equations:

∑j=1np {[(d~11Nij(3)+d~12Nij(2)+d~13Nij(1)+d~14Nij)(um^n^)j+(d~11N^ij(3)+d~12N^ij(2)+d~13N^ij(1)+d~14N^ij) (θum^n^)j+(d~11N¯ij(3)+d~12N¯ij(2)+d~13N¯ij(1)+d~14N¯ij)(κum^n^)j]+[(d~15Nij(1)+d~16Nij)(vm^n^)j+(d~15N^ij(1)+d~16N^ij)(θvm^n^)j+(d~15N¯ij(1)+d~16N¯ij)(κvm^n^)j]+[(d~17Nij(2)+d~18Nij(1)+d~19Nij)(wm^n^)j+(d~17N^ij(2)+d~18N^ij(1)+d~19N^ij)(θwm^n^)j+(d~17N¯ij(2)+d~18N¯ij(1)+d~19N¯ij)(κwm^n^)j]+[d~110Nij(σ31m^n^)j+d~110N^ij(θσ31m^n^)j+d~110N¯ij(κσ31m^n^)j]}=0,(59)

∑j=1np {[(d~21Nij(1)+d~22Nij)(um^n^)j+(d~21N^ij(1)+d~22N^ij)(θum^n^)j+(d~21N¯ij(1)+d~22N¯ij)(κum^n^)j]+[(d~23Nij(3)+d~24Nij(2)+d~25Nij(1)+d~26Nij)(vm^n^)j+(d~23N^ij(3)+d~24N^ij(2)+d~25N^ij(1)+d~26N^ij)(θvm^n^)j+(d~23N¯ij(3)+d~24N¯ij(2)+d~25N¯ij(1)+d~26N¯ij)(κvm^n^)j]+[(d~27Nij(2)+d~28Nij(1)+d~29Nij)(wm^n^)j+(d~27N^ij(2)+d~28N^ij(1)+d~29N^ij)(θwm^n^)j++(d~27N¯ij(2)+d~28N¯ij(1)+d~29N¯ij)(κwm^n^)j]+[d~210Nij(σ32m^n^)j+d~210N^ij(θσ32m^n^)j+d~210N¯ij(κσ32m^n^)]}=0,(60)

∑j=1np{[d~31Nij(um^n^)j+d~31N^ij(θum^n^)j+d~31N¯ij(κum^n^)j]+[d~32Nij(vm^n^)j+d~32N^ij(θvm^n^)j+d~32N¯ij(κvm^n^)j]−[Nij(1)(wm^n^)j+N^ij(1)(θwm^n^)j+N¯ij(1)(κwm^n^)j]+[d~33Nij(σ33m^n^)j+d~33N^ij(θσ33m^n^)j+d~33N¯ij(κσ33m^n^)j]}=0,(61)

∑j=1np {[(d~41Nij(2)+d~42Nij)(um^n^)j+(d~41N^ij(2)+d~42N^ij)(θum^n^)j+(d~41N¯ij(2)+d~42N¯ij)(κum^n^)j]+[(d~43Nij(2)+d~44Nij)(vm^n^)j+(d~43N^ij(2)+d~44N^ij)(θvm^n^)j+(d~43N¯ij(2)+d~44N¯ij)(κvm^n^)j]+[d~45Nij(1)(wm^n^)j+d~45N^ij(1)(θwm^n^)j+d~45N¯ij(1)(κwm^n^)j]−[Nij(1)(σ31m^n^)j+N^ij(1)(θσ31m^n^)j+N¯ij(1)(κσ31m^n^)j]+[d~46Nij(σ33m^n^)j+d~46N^ij(θσ33m^n^)j+d~46N¯ij(κσ33m^n^)j]}=0,(62)

∑j=1np {[(d~51Nij(2)+d~52Nij)(um^n^)j+(d~51N^ij(2)+d~52N^ij)(θum^n^)j+(d~51N¯ij(2)+d~52N¯ij)(κum^n^)j]+[(d~53Nij(2)+d~54Nij) (vm^n^)j+(d~53N^ij(2)+d~54N^ij) (θvm^n^)j+(d~53N¯ij(2)+d~54N¯ij) (κvm^n^)j]+[d~55Nij(1)(wm^n^)j+d~55N^ij(1)(θwm^n^)j+d~55N¯ij(1)(κwm^n^)j]−[Nij(1)(σ32m^n^)j+N^ij(1)(θσ32m^n^)j+N¯ij(1)(κσ32m^n^)j]+[d~56Nij(σ33m^n^)j+d~56N^ij(θσ33m^n^)j+d~56N¯ij(κσ33m^n^)j]}=0,(63)

∑j=1np {[(d~61Nij(3)+d~62Nij(2)+d~63Nij(1)+d~64Nij)(um^n^)j+(d~61N^ij(3)+d~62N^ij(2)+d~63N^ij(1)+d~64N^ij)(θum^n^)j+(d~61N¯ij(3)+d~62N¯ij(2)+d~63N¯ij(1)+d~64N¯ij)(κum^n^)j]+[(d~65Nij(3)+d~66Nij(2)+d~67Nij(1)+d~68Nij)(vm^n^)j+(d~65N^ij(3)+d~66N^ij(2)+d~67N^ij(1)+d~68N^ij)(θvm^n^)j+(d~65N¯ij(3)+d~66N¯ij(2)+d~67N¯ij(1)+d~68N¯ij)(κvm^n^)j]+[(d~69Nij(2)+d~610Nij(1)+d~611Nij)(wm^n^)j+(d~69N^ij(2)+d~610N^ij(1)+d~611N^ij)(θwm^n^)j+(d~69N¯ij(2)+d~610N¯ij(1)+d~611N¯ij)(κwm^n^)j]+[m~Nij(σ31m^n^)j+m~N^ij(θσ31m^n^)j+m~N¯ij(κσ31m^n^)j]+[n~Nij(σ32m^n^)j+n~N^ij(θσ32m^n^)j+n~N¯ij(κσ32m^n^)j]−[Nij(1)(σ33m^n^)j+N^ij(1)(θσ33m^n^)j+N¯ij(1)(κσ33m^n^)j]}=0,(64)

where i=1,2,…,nc; and the relevant coefficients are given in Appendix B.

The associated surface conditions, which are five conditions on the top surface and five conditions on the bottom surface, are given as

(σ31m^n^)1=(σ32m^n^)1=(σ33m^n^)1=(μxm^n^)1=(μym^n^)1=0when z=−h/2,(65a)

(σ31m^n^)nc=(σ32m^n^)nc=(μxm^n^)nc=(μym^n^)nc=0   and(σ33m^n^)nc=q¯m^n^when z=h/2.(65b)

As mentioned above, Eqs. (59)–(64) and (65a), (65b) represent a system of (6nc + 10) (i.e., 18np + 10) algebraic equations in terms of 18np nodal primary variables, which can be readily solved employing the weighted least square method.

4  Numerical Examples

4.1 Validation and Comparison Studies

This section considers a simply supported FG microplate that is subjected to either a sinusoidally distributed load or a uniform load. The former loading conditions are shown in Fig. 3.

The microplate of interest is made of alumina (Al2O3, a ceramic material) and aluminum (Al, a metal material). The microplate’s material properties are assumed to obey the power-law distributions for the constituents’ volume fractions, which vary in the thickness direction and are defined as follows:

Γcer(z)=[(1/2)+(z/h)] κp,(66a)

and

Γmet(z)=1−Γc(z),(66b)

where the subscripts cer and met denote the ceramic and metal materials, respectively.

The material properties of the alumina and the aluminum are given in the following form [23]:

For alumina material, Ecer=380  GPa,υcer=0.3,and ρcer=3800  kg/m3.(67a)

For aluminum material,

Emet=70  GPa, υmet=0.3, and ρmet=2702  kg/m3.(67b)

By using the rule of mixtures, we estimate the microplate’s effective material properties as follows:

Eeff(z)=Ecer Γcer(z)+Emet Γmet(z)=Emet+(Ecer−Emet) Γcer(z),(68a)

υeff(z)=0.3.(68b)

For comparison purposes, we define the non-dimensional variables in the same way as those used in Thai et al. [23]:

(u¯,  w¯)=(ux,  uζ) [10Ecer h3/(q0Lx4)],(69a)

σ¯ij=σij h/(q0Lx)(i, j=x, y, and z),except  σ¯ζζ=σζζ/q0.(69b)

When considering a homogeneous isotropic microplate, we change Ecer in Eq. (69a) to the microplate’s Young’s modulus, E0.

According to Lam et al.’s experimental results [7], this work defines the material length-scale parameters of the MCST and the CCST, l^ and l, respectively, as l^=2l=17.6×10−6 m. This is because the couple stress tensor (mij)-the curvature tensor (χij) relationship in the MCST is mij=2G l^2χij; however, the couple stress tensor (μij)-the curvature tensor (κij) relationship in CCST is μij=−8G l2κij. Thus, the relationship l^=2l is obtained, which can be employed to carry out a comprehensive comparison between the solutions obtained using the MCST and the CCST.

Table 1 shows the results of the Hermitian C2 DRK meshless method for the central deflection (i.e., w¯(Lx/2,Ly/2,0)) of a homogeneous microplate that is placed under full simple supports and is subjected to a sinusoidally distributed load, i.e., q¯z+(x,  y)=q0sin⁡(πx/Lx) sin⁡(πy/Ly) and q¯z−(x,  y)=0. The relevant material parameters are l/h=0,0.1,0.2,0.3,0.4,  and  0.5. The relevant geometric parameters are Lx=Ly and Lx/h=5. In Table 1, there are three types of sampling node distributions, which are Types A, B, and C, used with the total number of sampling nodes (np) being np = 13, 17, and 21, and with the base functions’ highest order being n = 4 and 5. In the case of Type A, the sampling nodes are uniformly distributed. In the case of Type B, distributions of the sampling nodes are selected using a formula of roots of the Chebyshev polynomial, which is ξi=−cos⁡[(i−1)π/(np−1)], where i=1,2,⋯, np. In the case of Type C, the sampling nodes are randomly scattered and have coordinates that are randomly generated by the computer we used. The sampling node distributions of Types A, B, and C are shown in Table 2.

Compared with the 3D solutions [50,51], Table 1 shows that the solutions obtained using our Hermitian C2 DRKIM method with n = 5 are more accurate than those with n = 4. The solutions obtained using the Hermitian C2 DRKIM method with the sampling node distributions of Types A and B are more precise than those with Type C sampling node distributions. For a range of the values of the l/h ratio from l/h = 0 to l/h = 0.5, the maximum relative error between the solutions obtained using the Hermitian C2 DRKIM method and those obtained using the 3D MCST [50] is 0.35% and 0.67% for Type A and Type B sampling node distributions, respectively. The relative error between the solutions obtained using 3D MCST and the 2D RSDT [23] is 2.3% in the case of l/h = 0, and it increases up to 17.1% in the case of l/h = 0.5. This is because the 3D couple stress effect is significant when the value of the l/h ratio increases. Because our Hermitian C2 DRKIM method is based on the strong form of the 3D CCST, its performance is superior to that of the 2D MCST-based microplate theory, especially for the microplates with a higher value of the l/h ratio.

Table 3 shows accuracy studies for the central deflection results of a simply supported homogeneous microplate obtained using our Hermitian C2 DRKIM method with a uniform node distribution (i.e., Type A) and different values of the support size al and the l/h ratio. The microplate considered here is subjected to the same loads as used in Table 1. The relevant parameters are l/h=0,0.1,0.2,0.3,0.4,  and  0.5; Lx=Ly and Lx/h=5; n = 5, np = 21. It can be seen in Table 3 that the solutions obtained using the current DRKIM method with the support sizes 3.5Δz≤al≤6.5Δz closely agree with the 3D MCST results [50] and the CCST-based FLM results [51]. The relative errors between the solutions obtained using the current DRKIM method and the relevant 3D solutions increase when the support size is smaller than al=3.5Δz and is larger than al=6.5Δz. Among these values of the support size al considered in Table 3, the selection of al=5.1Δz leads to a satisfactory result through the range of the l/h from l/h = 0 to l/h = 0.5, which is consistent with the guidance recommended by Chen et al. [45] and Wang et al. [46].

Table 4 shows that the comparisons of the central deflection results of a simply-supported FG microplate obtained using our Hermitian C2 DRKIM method with np = 31 and n = 5, the 3D CCST-based FEM [51], the refined quasi-3D IGAT [39], the MCST- and CCST-based RSDTs [23,27], and the CCST-based CPT [27]. The relevant geometric parameters are given as Lx/h = 5 and 20 and Lx=Ly. The relevant material parameters are l^/h=0,0.2,0.4,  and  0.8 and κp=0,1,  and  10.

The loading conditions considered here are sinusoidally distributed loads and uniform loads and are expressed as follows:

For the sinusoidal type load,

q¯z+(x,  y)=q0sin⁡(πx/Lx) sin⁡(πy/Ly),(70a)

q¯z−(x,  y)=0;(70b)

For the uniform-type load,

q¯z+(x,  y)=q0=∑m^=1,3,…nm^∑n^=1,3,…nn^16q0/(m^n^π2),(71a)

and q¯z−(x,  y)=0,(71b)

where in the following analysis, the convergent solutions are yielded when the values of nm^ and nn^ are taken to be nm^=nn^=29.

As previously mentioned, the microplate is made of Al2O3 and Al materials, and their volume fractions are given in Eqs. (66a) and (66b). Table 4 shows the Hermitian C2 DRKIM method’s results closely agree with those obtained using the 3D CCST-based FEM [51] and the quasi-3D IGAT [39]. It can be seen in Table 4 that the FG microplate’s central deflection decreases when the value of the material length-scale parameter rises, indicating that an increase in the material length-scale parameter stiffens the microscale plate, decreasing its central deflection. The central deflection increases when the value of the inhomogeneity index (κp) rises, indicating that an increase in the value κp softens the microplate because the stiffer ceramic material’s volume fraction will decrease as compared to the softer metal material’s volume fraction, increasing the microplate’s central deflection.

Table 5 shows the comparisons for the results of the FG microplate’s in-plane stresses and deflections obtained using various 2D microstructure-dependent shear deformation theories on the basis of the MCST/CCST and our Hermitian C2 DRKIM method. The relevant material parameters are κp=1  and  10; and l^/h=0,0.2,0.5,  and  1. The relevant geometric parameters are Lx/h = 10 and Lx=Ly. It can be seen in Table 5 that the solutions obtained using our Hermitian C2 DRKIM method closely agree with those obtained using the 3D CCST-based FEM [51] and are more accurate than those of 2D microstructure-dependent shear deformation theories. In addition, the in-plane stress solutions decrease when the value of l rises, indicating that an increase in the value of l stiffens the microplate, resulting in fewer deformations and fewer in-plane stresses induced in the microplate. Due to its excellent performance, we apply the Hermitian C2 DRKIM method to the following parametric study.

4.2 Parametric Study

This section presents a parametric study to understand the impact of essential factors on deformations, in-plane stresses, transverse stresses, and couple stresses induced in an EG microplate, which is placed under full simple supports and is subjected to mechanical loads. The microplate considered here is subjected to the sinusoidally distributed loads, which are q¯z+=q0sin⁡(πx/Lx) sin⁡(πy/Ly) and q¯z−=0. Material properties of the microplate are assumed to obey an exponential law, exponentially varying in the thickness direction as follows:

E(z)=Eb eκe[1/2+(z/h)],(72a)

υ(z)=0.3,(72b)

where Eb = 70 Gpa. When z=h/2, Eq. (72a) leads to Et=Eb eκe and κe=ln κ^e, in which κ^e=Et/Eb. The symbols κ^e and κe are defined as the inhomogeneity index and its logarithm form. The subscripts b and t stand for the bottom and top surfaces of the microplate. When the value of κ^e is one (i.e., the value of κe is zero), the FG microplate is a homogeneous plate.

The dimensionless variables used in the following study are given as

(u¯,  w¯)=(ux,  uζ) [10Eb h3/(q0Lx4)];(73a)

σ¯ij=σij h/(q0Lx)    i,j=x,  y,  and  ζ,except  σ¯ζζ=σζζ/q0;(73b)

(μ¯x,  μ¯y,  μ¯ζ)=[μx/(q0Lx),  μy/(q0Lx),  μζ/(q0h)].(73c)

It is important to note that using dimensionless variables in the following analysis allows for a more comprehensive understanding of the results. For instance, when the value of the l/h ratio is fixed, like l/h = d, as we change the value of l and let h = l/d, or we change the value of h and let l = (hd), we always obtain the same results. This approach enhances the robustness and applicability of our findings.

Fig. 4a–h shows the variations in dimensionless displacements, in-plane stresses, transverse stresses, and couple stresses induced in an EG microplate along the thickness direction, with the values of the inhomogeneity index (κ^e) being 1, 5, and 10. The relevant material parameters are l/h=0.5 and l=8.8×10−6 m. The relevant geometric parameters are Lx/h=10 and Lx/Ly=1. The results in Fig. 4b show that the microplate’s overall stiffness increases when the value of κ^e rises, decreasing the microplate’s deflection. The results in Fig. 4c and d show that the in-plane normal and shear stress distributions along the thickness direction look like higher-order polynomial functions for an EG microplate (κ^e≠1); however, these distributions look like linear functions for a homogeneous microplate (κ^e=1). Fig. 4e and f shows that the transverse shear and normal stress distributions along the thickness direction look like higher-order polynomial functions in an EG microplate, and the pick value occurs in the upper half of the microplate; however, these distributions appear to be parabolic functions in a homogeneous microplate with the pick value occurring at the microplate’s mid-plane. Fig. 4g and h shows that the couple stress distribution along the thickness direction looks like higher-order polynomial functions. The variations in the induced stress and deformation distributions along the thickness direction for an FG microplate are more significant than those for a homogeneous microplate.

Figure 4: Variations in the dimensionless (a) in-plane displacement u¯, (b) out-of-plane displacement w¯, (c) in-plane normal stress σ¯xx, (d) in-plane shear stress σ¯xy, (e) transverse shear stress σ¯ζx, (f) transverse normal stress σ¯ζζ, (g) couple stress μ¯x, and (h) couple stress μ¯ζ along the thickness direction, with the value of the inhomogeneity index being 1, 5, and 10

Fig. 5a–h shows the variations in the dimensionless displacement, in-plane stresses, transverse stresses, and couple stresses induced in an EG microplate along the thickness direction, with the values of the l/h ratio being 0, 0.2, and 0.4. The other material parameter is κe=2; and the relevant geometric parameters are Lx/Ly=1, Lx/h=10, and h=1×10−6 m.

Figure 5: Variations in the dimensionless (a) in-plane displacement u¯, (b) out-of-plane displacement w¯, (c) in-plane normal stress σ¯xx, (d) in-plane shear stress σ¯xy, (e) transverse shear stress σ¯ζx, (f) transverse normal stress σ¯ζζ, (g) couple stress μ¯x, and (h) couple stress μ¯ζ along the thickness direction, with the value of the l/h ratio being 0, 0.2, and 0.4

It can be seen in Fig. 5b that an increase in the value of l stiffens the microplate, leading to its deflection decrease. The results in Fig. 5c–f show that the variations in the in-plane stress and transverse stress induced in the microplate along its thickness direction with a smaller value of l are more significant than those induced in the microplate along the thickness direction with a more considerable value of l. This is because an increase in the value of l stiffens the microplate, which leads to fewer deformations and fewer stresses induced in the microplate when the magnitude of the applied load remains constant. The results in Fig. 5g and h show that the variations in the couple stresses (μ¯x  and  μ¯ζ) along the thickness coordinate look like higher-order polynomial functions.

Fig. 6a–h shows that the variations in the dimensionless deformations, in-plane stresses, transverse stresses, and couple stresses induced in an EG microplate along the thickness direction, with the length-to-thickness ratios being 5, 10, and 20. The relevant material parameters are l/h = 0.5 and l=8.8×10−6 m. The dimensionless displacements are redefined as (u¯,  w¯)=(ux,  uζ) [10Eb/(q0h)]. The results in Fig. 6a and b show that the microplate’s overall stiffness decreases when the value of the Lx/h ratio rises, increasing its deflection. The results in Fig. 6c–f show that the variations in the in-plane stresses and transverse stresses induced in the microplate along the thickness direction with a more considerable value of the Lx/h ratio are more significant than those induced in the microplate along the thickness direction with a smaller value of the Lx/h ratio. This is because a decrease in the value of the Lx/h ratio stiffens the microplate, leading to fewer deformations and fewer stresses induced in the microplate when the magnitude of the applied load remains constant.

Figure 6: Variations in the dimensionless (a) in-plane displacement u¯, (b) out-of-plane displacement w¯, (c) in-plane normal stress σ¯xx, (d) in-plane shear stress σ¯xy, (e) transverse shear stress σ¯ζx, (f) transverse normal stress σ¯ζζ, (g) couple stress μ¯x, and (h) couple stress μ¯ζ along the thickness direction, with the values of the Lx/h ratio being 5, 10, and 20

5  Concluding Remarks

A Hermitian C2 DRKIM method based on the strong form of the 3D CCST has been developed for analyzing the 3D microstructure-dependent static flexural behavior of an FG/EG microplate. The FG/EG microplate considered here was subjected to mechanical loads and placed under full simple supports. The unique features of this Hermitian C2 DRK interpolant compared with the early Lagrange-type reproducing kernel interpolant for analyzing macroscale plate’s mechanical behavior are that the displacements, the transverse stresses, and their first-order and second-order derivatives are selected as primary variables satisfying the nodal interpolation properties, and their corresponding shape functions satisfy the Kronecker delta properties. These features make our Hermitian C2 DRKIM method suitable for analyzing the FG microplate’s mechanical behavior because the deflections and rotations prescribed at the boundary edges of the microplate considered here can thus be directly imposed without using the penalty method, which is necessary for the conventional reproducing kernel point collocation method. In addition, using our Hermitian C2 DRKIM method, the primary variables’ higher-order derivatives involved in the strong form of the CCST can be effectively estimated.

In the validation and comparison study, the solutions obtained using our Hermitian C2 DRKIM method closely agree with the available 3D solutions in the literature, with a fast convergence rate. Because the 3D couple stress effect is significant when the value of the l/h ratio rises, the performance of the Hermitian C2 DRKIM method is superior to that of 2D CCST-/MCST-based shear deformation microplate theories, especially for the microplate with a considerable value of the l/h ratio. For example, the maximum relative error between the solutions obtained using the Hermitian C2 DRKIM method and those obtained using the 3D MCST is 0.35% for a range of the value of the l/h ratio between l/h = 0 and l/h = 0.5, respectively; however, the relative error between the solutions obtained using the 2D RSDT and the 3D MCST solutions is 2.3% in the case of l/h = 0, and it increases up to 17.1% when the value of the l/h ratio is 0.5.

In the parametric study, we presented the displacement, in-plane stress, transverse stress, and couple stress distributions along the thickness direction of an EG microplate using our Hermitian C2 DRKIM method. These distributions cannot be effectively estimated using existing 2D microstructure-dependent shear deformation theories, especially for the transverse stress and couple stress distributions, so they have yet to be shown in public literature. Thus, the parametric analysis results can provide a reference for assessing the accuracy of existing 2D microstructure-dependent shear deformation theories. Furthermore, the results are also helpful for making assumptions about primary variable components for an advanced microstructure-dependent shear deformation microplate theory, which is to be developed.

Acknowledgement: The authors thank the National Science and Technology Council of the Republic of China for its financial support.

Funding Statement: This study was supported by a grant from the National Science and Technology Council of the Republic of China (Grant Number: MOST 112-2221-E-006-048-MY2).

Author Contributions: Chih-Ping Wu: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Resources, Data curation, Writing—original draft preparation, Writing—review and editing, Supervision, Project administration, Funding acquisition. Ruei-Syuan Chang: Software, Validation, Formal analysis, Investigation, Data curation. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: The processed data required to reproduce these findings can be downloaded from https://docs.google.com/document/d/1hvfVYcF5hsGj-4HEn4WOJL9y3hNq68Q/edit?usp=drive_link&ouid=103032847656566806520&rtpof=true&sd=true (accessed on 20 May 2024).

Conflicts of Interest: The authors declare they have no conflicts of interest to report regarding the current study.

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Appendix A: The Higher-Order Derivative of the Hermitian C2 DRK Interpolant

The rth-order derivative of the Hermitian C2 DRK interpolant fh (ξ) with respect to ξ is expressed as follows:

drfh(ξ)dξr=∑l=1np  [Nl(r)(ξ)fl+N^l(r)(ξ)θl+N¯l(r)(ξ)κl]=∑l=1np [ (ϕl(r)(ξ)+ψl(r)(ξ))  fl+(ϕ^l(r)(ξ)+ψ^l(r)(ξ))  θl+(ϕ¯l(r)(ξ)+ψ¯l(r)(ξ))  κl] ,  (74)

where Nl(r)(ξ), N^l(r)(ξ), and N¯l(r)(ξ) are the shape functions of Hermitian C2 DRK interpolant’s rth-order derivatives at the sampling node ξ = ξl; ψl(r)(ξ), ψ^l(r)(ξ), and ψ¯l(r)(ξ) (l = 1, 2,…, np) denote primitive functions’ rth-order derivatives (i.e., ψl(r)(ξ)=dr ψl(ξ)/dξr, ψ^l(r)(ξ)=dr ψ^l(ξ)/dξr, and ψ¯l(r)(ξ)=dr ψ¯l(ξ)/dξr); and ϕl(r)(ξ), ϕ^l(r)(ξ), and ϕ¯l(r)(ξ) (l = 1, 2,…, np) denote enrichment functions’ rth-order derivatives, which are obtained by imposing the nth-order DRCs, and are given by ϕl(r)(ξ)=wa(ξ−ξl)PT(ξ−ξl) brc2(ξ), ϕ^l(r)(ξ)=wa(ξ−ξl)P^T(ξ−ξl) brc2(ξ), and ϕ¯l(r)(ξ)=wa(ξ−ξl)P¯T(ξ−ξl) brc2(ξ), in which brc2(ξ) is the undetermined function vector.

The undetermined functions brc2(ξ) in Eq. (74) can be determined by selecting the complete nth-order polynomials as the basis functions to be reproduced and set up (n + 1) DRCs as follows:

∑l=1np P(ξ−ξl)  ϕ l(r) (ξ)+∑l=1np P^(ξ−ξl)  ϕ^l(r) (ξ)+∑l=1np P¯(ξ−ξl)  ϕ¯l(r) (ξ)=drP(0)/dξlr−∑l=1np P(ξ−ξl)  ψl(r) (ξ)−∑l=1np P^(ξ−ξl)  ψ^l(r) (ξ)−∑l=1np P¯(ξ−ξl)  ψ¯l(r) (ξ).(75)

We substitute the enrichment functions in Eq. (74) into the differential reproducing conditions in Eq. (75). As a result, we obtain the undetermined function vector b rc2(ξ) as follows:

b rc2(ξ)=Ac2−1(ξ)[drP(0)/dξlr−∑l=1np P(ξ−ξl)  ψl(r) (ξ)−∑l=1np P^(ξ−ξl)  ψ^l(r) (ξ)−∑l=1np P¯(ξ−ξl)  ψ¯l(r) (ξ)].(76)

We substitute Eq. (76) into Eq. (74) to obtain the shape functions of Hermitian C2 DRK interpolant’s rth-order derivatives as follows:

Nl(r)(ξ)=ϕ l (r)(ξ)+ψ l (r) (ξ)(l=1,2,…,np),(77)

N^l(r)(ξ)=ϕ^l (r)(ξ)+ψ^l (r) (ξ)(l=1,2,…,np),(78)

N¯l(r)(ξ)=ϕ¯l (r)(ξ)+ψ¯l (r) (ξ)(l=1,2,…,np),(79)

where

ϕ l(r)(ξ)=wa(ξ−ξl)PT(ξ−ξl)brc2(ξ), ϕ^l(r)(ξ)=wa(ξ−ξl)P^T(ξ−ξl)brc2(ξ), ϕ¯l(r)(ξ)=wa(ξ−ξl)P¯T(ξ−ξl)brc2(ξ).

Appendix B: The Relevant Coefficients d~ij

The relevant coefficients d~ij are given as follows:

d~11=G32c55−1l2, d~12=G32,z c55−1l2, d~13=−(m~2G21+n~2G32)c55−1l2−1, d~14=−n~2G32,z c55−1l2,

d~15=m~n~(G32−G21)c55−1l2, d~16=m~n~G32,z c55−1l2, d~17=−m~G32c55−1l2, d~18=−m~G32,z c55−1l2,

d~19=(m~3+m~n~2)G21c55−1l2−m~, d~110=c55−1, d~21=m~n~(G13−G21)c44−1l2, d~22=m~n~G13,z c44−1l2,

d~23=G13c44−1l2, d~24=G13,z c44−1l2, d~25=−(m~2G13c44−1+n~2G21c44−1)l2−1, d~26=−m~2G13,z c44−1l2, d~27=−n~G13c44−1l2, d~28=−n~G13,z c44−1l2, d~29=(m~2n~+n~3)G21c44−1l2−n~, d~210=c44−1, d~31=m~c~13,

d~32=n~c~23, d~33=c33−1, d~41=−n~2G32l2, d~42=(m~2Q11+n~2c66)+(m~2n~2G13+n~4G32)l2, d~43=m~n~G13l2,

d~44=m~n~(Q12+c66)−(m~3n~G13+m~n~3G32)l2,d~45=(−m~n~2G13+m~n~2G32)l2,d~46=−m~c~13,

d~51=m~n~G32l2, d~52=m~n~(Q12+c66)−(m~n~3G32+m~3n~G13)l2, d~53=−m~2G13l2,

d~54=(n~2Q22+m~2c66)+(m~2n~2G32+m~4G13)l2, d~55=(m~2n~G13−m~2n~G32)l2, d~56=−n~c~23,

d~61=2m~G32l2, d~62=2m~G32,z l2, d~63=[−2m~n~2G32+2m~n~2G13−2(m~3+m~n~2)G21]l2,

d~64=[−2m~n~2G32,z+2m~n~2G13,z] l2, d~65=2n~G13l2, d~66=2n~G13,z l2,

d~67=[2m~2n~G32−2m~2n~G13−2(m~2n~+n~3)G21]l2, d~68=[2m~2n~G32,z−2m~2n~G13,z] l2,

d~69=−2(m~2G32+n~2G13)l2, d~610=−2[m~2G32,z+n~2G13,z] l2, d~611=2(m~4+2m~2n~ 2+n~4)G21l2.

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